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Stochastic differential equations and their applications. (Stokhasticheskie differentsial’nye uravneniya i ikh prilozheniya). (Russian) Zbl 0557.60041

Akademiya Nauk Ukrainskoj SSR. Ordena Trudovogo Krasnogo Znameni Institut Matematiki. Kiev: ”Naukova Dumka”. 612 p. R. 5.70 (1982).
14 years have passed since the first comprehensive monograph ”Stochastic differential equations” was published by the present authors (see Zbl 0169.487). Since then the subject has considerably expanded by new branches such as the ”théorie générale”, the martingale problem approach - to name but a few. The present monograph gives a fairly self- contained development of the modern theory of stochastic differential equations on a Euclidean state space, and only a basic knowledge of probability theory and stochastic processes is assumed.
The contents are as follows: Chapter 1 provides the basic facts about martingales. First a detailed analysis of the discrete time case is given. Then continuous time martingales are analyzed. Chapter 2 contains that part of the general theory of processes which in the opinion of the authors is necessary for the following. It deals with the classification of processes and stopping times. Doleans measure and projections of processes onto the predictable and optional sets, spaces of martingales, the Doob-Meyer decomposition, and local characteristics. Chapter 3 is on stochastic integration with respect to martingales and random measures, the Itô formula and on changes of measure. Chapter 4 deals with strong solutions of stochastic differential equations. In the first two sections the equations are driven by independent increment processes, resp. by martingales and random measures. In the third section the equation (as in the first author’s original approach) is driven by a random vector field. Chapter 5 deals with weak solutions and limit theorems, and chapter 6 describes the relation between stochastic differential equations and Markov processes. Finally, the historical development of the subject of each chapter is outlined at the end of the book.
Rather than describing the contents of this book in more detail we shall give a few examples of what it does not contain: 1.) By its restriction to the Euclidean state space it does not deal with stochastic differential equations in infinite dimensions or on manifolds. 2.) It does not treat the general problem of stochastic integration and its recent solution (that semimartingales are the only ”good” stochastic integrators in finite dimensions). 3.) In spite of its title it does not deal with applications (outside mathematics). - Moreover, it does not contain a subject index.
Nevertheless, in many other aspects it is quite complete, it contains material which cannot be found in other books on this subject, and theorems or assumptions are often illustrated by examples. Therefore it would be a helpful source both for the advanced student and for the researcher if translated into English.
Reviewer: P.Kotelenez

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J25 Continuous-time Markov processes on general state spaces
60G42 Martingales with discrete parameter
60G07 General theory of stochastic processes
60G44 Martingales with continuous parameter

Citations:

Zbl 0169.487